The fitness level method is a popular tool for analyzing the computation time of elitist evolutionary algorithms. Its idea is to divide the search space into multiple fitness levels and estimate lower and upper bounds on the computation time using transition probabilities between fitness levels. However, the lower bound generated from this method is often not tight. To improve the lower bound, this paper rigorously studies an open question about the fitness level method: what are the tightest lower and upper time bounds that can be constructed based on fitness levels? To answer this question, drift analysis with fitness levels is developed, and the tightest bound problem is formulated as a constrained multi-objective optimization problem subject to fitness level constraints. The tightest metric bounds from fitness levels are constructed and proven for the first time. Then the metric bounds are converted into linear bounds, where existing linear bounds are special cases. This paper establishes a general framework that can cover various linear bounds from trivial to best coefficients. It is generic and promising, as it can be used not only to draw the same bounds as existing ones, but also to draw tighter bounds, especially on fitness landscapes where shortcuts exist. This is demonstrated in the case study of the (1+1) EA maximizing the TwoPath function.

翻译：适应度层方法是分析精英进化算法计算时间的常用工具。其核心思想是将搜索空间划分为多个适应度层，并通过层间转移概率估算计算时间的下界与上界。然而，该方法生成的下界通常不够紧致。为改进下界，本文严格研究了适应度层方法的一个开放性问题：基于适应度层可构造的最紧致下界与上界分别是什么？针对该问题，本文发展了适应度层漂移分析方法，并将最紧致界问题形式化为受适应度层约束的多目标优化问题。首次构建并证明了基于适应度层的最紧致度量界。随后将度量界转换为线性界，其中现有线性界均为特例。本文建立了一个通用框架，可涵盖从平凡系数到最优系数的各类线性界。该框架具有通用性和实用性，不仅能复现现有界，还能在存在捷径的适应度景观中构造更紧致的界，这在(1+1) EA最大化双峰函数的案例研究中得到验证。